Homogenization of Neumann problems for unbounded integral functionals
Bollettino della Unione matematica italiana, Série 8, 2B (1999) no. 2, pp. 463-491.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

Si studia l'omogeneizzazione di problemi di tipo Neumann per funzionali integrali del Calcolo delle Variazioni definiti su funzioni soggette a vincoli puntuali di tipo oscillante sul gradiente, in ipotesi minimali sui vincoli. I risultati ottenuti sono dedotti mediante l'introduzione di nuove tecniche di $\Gamma$-convergenza, unitamente ad un risultato di ricostruzione per funzioni affini a tratti, che permettono la dimostrazione di un teorema generale di omogeneizzazione per funzionali integrali a valori reali estesi.
@article{BUMI_1999_8_2B_2_a19,
     author = {Carbone, Luciano and Corbo Esposito, Antonio and De Arcangelis, Riccardo},
     title = {Homogenization of {Neumann} problems for unbounded integral functionals},
     journal = {Bollettino della Unione matematica italiana},
     pages = {463--491},
     publisher = {mathdoc},
     volume = {Ser. 8, 2B},
     number = {2},
     year = {1999},
     zbl = {0940.49015},
     mrnumber = {724197},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_1999_8_2B_2_a19/}
}
TY  - JOUR
AU  - Carbone, Luciano
AU  - Corbo Esposito, Antonio
AU  - De Arcangelis, Riccardo
TI  - Homogenization of Neumann problems for unbounded integral functionals
JO  - Bollettino della Unione matematica italiana
PY  - 1999
SP  - 463
EP  - 491
VL  - 2B
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_1999_8_2B_2_a19/
LA  - en
ID  - BUMI_1999_8_2B_2_a19
ER  - 
%0 Journal Article
%A Carbone, Luciano
%A Corbo Esposito, Antonio
%A De Arcangelis, Riccardo
%T Homogenization of Neumann problems for unbounded integral functionals
%J Bollettino della Unione matematica italiana
%D 1999
%P 463-491
%V 2B
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_1999_8_2B_2_a19/
%G en
%F BUMI_1999_8_2B_2_a19
Carbone, Luciano; Corbo Esposito, Antonio; De Arcangelis, Riccardo. Homogenization of Neumann problems for unbounded integral functionals. Bollettino della Unione matematica italiana, Série 8, 2B (1999) no. 2, pp. 463-491. http://geodesic.mathdoc.fr/item/BUMI_1999_8_2B_2_a19/

[An] B. D. Annin , Existence and uniqueness of the solution of the elastic-plastic torsion problem for a cylindrical bar of oval cross-section, J. Appl. Math. Mech., 29 (1965), 1038-1047. | Zbl

[A1] H. Attouch , Introduction a l'homogenisation d'inequations variationnelles, Rend. Sem. Mat. Univ. Politecn. Torino, 40 (1982), 1-23. | MR | Zbl

[A2] H. Attouch , Variational Convergence for Functions and Operators, Pitman (1984). | MR | Zbl

[BLP] A. Bensoussan - J. L. Lions - G. Papanicolaou , Asymptotic Analysis for Periodic Structures, Stud. Math. Appl., 5, North Holland (1978). | MR | Zbl

[B] H. Brezis , Multiplicateur de Lagrange en torsion elasto-plastique, Arch. Rational Mech. Anal., 49 (1973), 32-40. | MR | Zbl

[BS] H. Brezis - M. Sibony , Équivalence de deux inéquations variationnelles et applications, Arch. Rational Mech. Anal., 41 (1971), 254-265. | MR | Zbl

[Bu] G. Buttazzo , Su una definizione generale dei $\Gamma$-limiti, Boll. Un. Mat. Ital. (5), 14-B (1977), 722-744. | MR | Zbl

[CR] A. Caffarelli - N. M. Riviere , On the Lipschitz character of the stress tensor when twisting an elastic-plastic bar, Arch. Rational Mech. Anal., 69 (1979), 31-36. | MR | Zbl

[C1] L. Carbone , Sur la convergence des intégrales du type de l'énergie sur des fonctions a gradient borné, J. Math. Pures Appl., 56 (1977), 79-84. | MR | Zbl

[C2] L. Carbone , $\Gamma$-convergence d'intégrales sur des functions avec des contraintes sur le gradient, Comm. Partial Differential Equations, 2 (1977), 627-651. | MR | Zbl

[C3] L. Carbone , Sull'omogeneizzazione di un problema variazionale con vincoli sul gradiente, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8), Mat. Appl., 63 (1977), 10-14. | Zbl

[C4] L. Carbone , Sur un problème d'homogénéisation avec des contraintes sur le gradient, J. Math. Pures Appl., 58 (1979), 275-297. | MR | Zbl

[CDA1] L. Carbone - R. De Arcangelis , On integral representation, relaxation and homogenization for unbounded functionals, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9), Mat. Appl., 8 (1997), 129-135. | MR | Zbl

[CDA2] L. Carbone - R. De Arcangelis , Unbounded functionals: applications to the homogenization of gradient constrained problems, to appear on Ricerche Mat.. | MR | Zbl

[CS1] L. Carbone - S. Salerno , On a problem of homogenization with quickly oscillating constraints on the gradient, J. Math. Anal. Appl., 90 (1982), 219-250. | MR | Zbl

[CS2] L. Carbone - S. Salerno , Further results on a problem of homogenization with constraints on the gradient, J. Analyse Math., 44 (1984/85), 1-20. | MR | Zbl

[CS3] L. Carbone - S. Salerno , Homogenization with unbounded constraints on the gradient, Nonlinear Anal., 9 (1985), 431-444. | MR | Zbl

[CS] L. Carbone - C. Sbordone , Some properties of $\Gamma$-limits of integral functionals, Ann. Mat. Pura Appl. (4), 122 (1979), 1-60. | MR | Zbl

[CSJP] D. Cioranescu - J. Saint Jean Paulin , Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607. | MR | Zbl

[CEDA1] A. Corbo Esposito - R. De Arcangelis , The Lavrentieff phenomenon and different processes of homogenization, Comm. Partial Differential Equations, 17 (1992), 1503-1538. | MR | Zbl

[CEDA2] A. Corbo Esposito - R. De Arcangelis , Homogenization of Dirichlet problems with nonnegative bounded constraints on the gradient, J. Analyse Math., 64 (1994), 53-96. | MR | Zbl

[CEDA3] A. Corbo Esposito - R. De Arcangelis , Some notes on a characterization of function sets described by constraints on the gradient, in Nonlinear Variational Problems and Partial Differential Equations, A. Marino and M.K.V. Murthy editors, Pitman Res. Notes Math. Series, 320, Longman Scientific & Technical (1995), 182-196. | Zbl

[CESC] A. Corbo Esposito - F. Serra Cassano , A Lavrentieff phenomenon for problems of homogenization with constraints on the gradient, Ricerche Mat., 46 (1997), 127-159. | MR | Zbl

[DM] G. Dal Maso , An Introduction to $\Gamma$-Convergence, Progr. Nonlinear Differential Equations Appl., 8, Birkhäuser (1993). | MR | Zbl

[DAG] R. De Arcangelis - G. Gargiulo , Homogenization of integral functionals with linear growth defined on vector-valued functions, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 371-416. | MR | Zbl

[DAGP] R. De Arcangelis - A. Gaudiello - G. Paderni , Some cases of homogenization of linearly coercive gradient constrained variational problems, Math. Models Methods Appl. Sci., 6 (1996), 901-940. | MR | Zbl

[DAV] R. De Arcangelis - A. Vitolo , Some cases of homogenization with unbounded oscillating constraints on the gradient, Asymptotic Anal., 5 (1992), 397-428. | MR | Zbl

[DG] E. De Giorgi , $G$-operators and $\Gamma$-convergence; in Proceedings of the International Congress of Mathematicians, Warszawa, August 16-24, 1983, PWN, Warszawa (1984), 1175-1191. | Zbl

[DGF] E. De Giorgi - T. Franzoni , Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia, 3 (1979), 63-101. | Zbl

[DGS] E. De Giorgi - S. Spagnolo , Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4), 8 (1973), 391-411. | MR | Zbl

[DMD] U. De Maio - T. Durante , Homogenization of Dirichlet problems for some types of integral functionals, to appear on Ricerche Mat. | MR | Zbl

[DL] G. Duvaut - H. Lanchon , Sur la solution du problème de torsion élastoplastique d'une barre cylindrique de section quelconque, C. R. Acad. Sci. Paris Sér. I Math., 264 (1967), 520-523. | MR | Zbl

[DLi] G. Duvaut - J.L. Lions , Inequalities in Mechanics and Physics, Springer-Verlag (1976). | MR | Zbl

[F] N. Fusco , $\Gamma$-convergenza unidimensionale, Boll. Un. Mat. Ital. (5), 16-B (1979), 74-86. | MR | Zbl

[GL] R. Glowinski - H. Lanchon , Torsion élastoplastique d'une barre cylindrique de section multiconnexe, J. Mécanique, 12 (1973), 151-171. | MR | Zbl

[L] H. Lanchon , Torsion élastoplastique d'une barre cylindrique de section simplement ou multiplement connexe, J. Mécanique, 13 (1974), 267-320. | MR | Zbl

[M] P. Marcellini , Periodic solutions and homogenization of non linear variational problems, Ann. Mat. Pura Appl. (4), 117 (1978), 139-152. | MR | Zbl

[MS] P. Marcellini - C. Sbordone , Homogenization of non-uniformly elliptic operators, Appl. Analysis, 8 (1978), 101-113. | MR | Zbl

[MT] F. Murat - L. Tartar , $H$-convergence, in Topics in the Mathematical Modelling of Composite Materials, A. Cherkaev and R. Kohn editors, Progr. Nonlinear Differential Equations Appl., 31, Birkäuser (1997), 21-44. | MR | Zbl

[SP] E. Sanchez.Palencia , Nonhomogeneous Media and Vibration Theory, Lecture Notes in Phys., 127, Springer-Verlag (1980). | MR | Zbl

[T] T. W. Ting , Elastic-plastic torsion of simply connected cylindrical bars, Indiana Univ. Math. J., 20 (1971), 1047-1076. | MR | Zbl

[ZKON] V. V. Zhikov - S. M. Kozlov - O. A. Oleinik - T'En Ngoan , Averaging and $G$-convergence of differential operators, Russian Math. Surveys, 34 (1979), 69-147. | MR | Zbl